A decoupling property of some Poisson structures on [math] supporting [math] Poisson–Lie symmetry

Journal of Mathematical Physics, Volume 62, Issue 3, March 2021. We study a holomorphic Poisson structure defined on the linear space [math] that is covariant under the natural left actions of the standard [math] and [math] Poisson–Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the d = 1 special case gives a Poisson structure on S(n, 1), and we construct a local Poisson map from the Cartesian product of d independent copies of S(n, 1) into S(n, d), which is a holomorphic diffeomorphism in a neighborhood of 0. The Poisson structure on S(n, d) is the complexification of a real Poisson structure on [math] constructed by the authors and Marshall, where a similar decoupling into d independent copies was observed. We also relate our construction to a Poisson structure on S(n, d) defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars–Schneider system by Hamiltonian reduction.